Activities This Week

Recently, Ofek, Pandey, and Shalit have defined a version of Banach-Mazur distances on the space of isomorphism classes of finite-dimensional complete Pick spaces. By the universality theorem of Agler and McCarthy, every finite-dimensional complete Pick space on n points is equivalent to a subspace of the Drury-Arveson space spanned by n kernels at points of the unit ball of some C^d. We propose to study the space of projections on finite-dimensional multiplier coinvariant subspaces of the Drury-Arveson space. The metric on this space is induced by the norm. We show that if we restrict ourselves to the subspace of projection on spaces spanned by distinct n kernels, then this space is homeomorphic to the symmetrized polyball. It then follows that the invariant distance obtained induces the same topology on the space of isomorphism classes of complete Pick space as the Banach-Mazur distance of Ofek, Pandey, and Shalit. Time permuting we will show a potential application of this idea

For a smooth projective hyperbolic curve Y/Q the set of rational points Y(Q) is finite by Faltings’ Theorem. Grothendieck’s section conjecture predicts that this set can be described via Galois sections of the étale fundamental group of Y. On the other hand, the non-abelian Chabauty method produces p-adic analytic functions which conjecturally cut out Y(Q) as a subset of Y(Qp). We relate the two conjectures and discuss the example of the thrice-punctured line, where non-abelian Chabauty is used to prove a local-to-glocal principle for the section conjecture.

We study the notion of Dirichlet improvability in a variety of settings and make a comparison study between Dirichlet-improvable numbers and badly-approximable numbers as initiated by Davenport-Schmidt. The question we try to answer, in each of the settings, is – whether the set of badly-approximable numbers is contained in the set of Dirichlet-improvable numbers. We show how this translates into a question about the possible limit points of bounded orbits in the space of two-dimensional lattices under the diagonal flow. Our main result gives a construction of a full Hausdorff dimension set of lattices with bounded orbit and with a prescribed limit point. Joint work with Dmitry Kleinbock.